Download Chapter 31

Chapter 31
Alternating Current
PowerPoint® Lectures for
University Physics, 14th Edition
– Hugh D. Young and Roger A. Freedman
© 2016 Pearson Education Inc.
Lectures by Jason Harlow
Learning Goals for Chapter 31
Looking forward at …
• How phasors make it easy to describe sinusoidally varying
quantities.
• How to use reactance to describe the voltage across a circuit
element that carries an alternating current.
• How to analyze an L-R-C series circuit with sinusoidal emfs
of different frequencies.
• What determines the amount of power flowing into or out of
an alternating current circuit.
• Why transformers are useful, and how they work.
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Introduction
• Waves from a broadcasting
station produce an alternating
current in the circuits of a
radio (like the one in this
classic car).
• How does a radio tune to a
particular station?
• How are ac circuits different from dc circuits?
• We shall see how resistors, capacitors, and inductors behave
with a sinusoidally varying voltage source.
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AC sources
• Most present-day household
and industrial power
distribution systems operate
with alternating current (ac).
• Any appliance that you plug
into a wall outlet uses ac.
• An ac source is a device that
supplies a sinusoidally
varying voltage.
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AC sources and currents
• A sinusoidal voltage might be described by a function
such as:
• Here v is the instantaneous potential difference, V is the
voltage amplitude, and ω = 2πf is the angular frequency.
• In the United States and Canada, commercial electric-power
distribution systems use a frequency f = 60 Hz.
• The corresponding sinusoidal alternating current is:
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Phasor diagrams
• To represent sinusoidally varying voltages and currents, we
define rotating vectors called phasors.
• Shown is a phasor diagram
for sinusoidal current.
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Root-mean-square values
• To calculate the rms value of
a sinusoidal current:
1. Graph current i versus
time.
2. Square the instantaneous
current i.
3. Take the average (mean)
value of i2.
4. Take the square root of that
average.
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Root-mean-square values
• For sinusoidal ac sources, the rms current and voltage values
are:
• This wall socket has a voltage amplitude of V = 170 V,
meaning that the voltage alternates
between +170 V and −170 V.
• The rms voltage is Vrms = 120 V.
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Resistor in an ac circuit: Slide 1 of 3
• When a resistor is
connected with an
ac source, the voltage
and current amplitudes
are related by
Ohm’s law:
• The resistance does not depend on the frequency of the
ac source.
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Resistor in an ac circuit: Slide 2 of 3
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Resistor in an ac circuit: Slide 3 of 3
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Inductor in an ac circuit: Slide 1 of 3
• When an inductor is
connected with an
ac source, the voltage
and current amplitudes
are related by:
• The inductive reactance is XL = ωL; the greater the
inductance and the higher the frequency, the greater the
inductive reactance.
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Inductor in an ac circuit: Slide 2 of 3
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Inductor in an ac circuit: Slide 3 of 3
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Capacitor in an ac circuit: Slide 1 of 3
• When a capacitor is
connected with an
ac source, the voltage
and current amplitudes
are related by:
• The capacitive reactance is XC = 1/ωC; the greater the
capacitance and the higher the frequency, the smaller the
capacitive reactance.
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Capacitor in an ac circuit: Slide 2 of 3
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Capacitor in an ac circuit: Slide 3 of 3
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Comparing ac circuit elements
• The graph shows how the resistance of a resistor and the
reactances of an inductor and a capacitor vary with angular
frequency ω.
• Resistance R is
independent of frequency.
• If ω = 0, corresponding to
a dc circuit, there is no
current through a capacitor
because XC → ∞.
• In the limit ω → ∞, the
current through an inductor
becomes vanishingly small.
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A useful application: The loudspeaker
• In order to route signals of
different frequency to the
appropriate speaker shown, the
woofer and tweeter are
connected in parallel across the
amplifier output.
• The capacitor in the tweeter
branch blocks the low-frequency
components of sound but passes
the higher frequencies.
• The inductor in the woofer
branch blocks the highfrequency components of sound
but passes the lower frequencies.
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The L-R-C series circuit: Slide 1 of 3
• When a resistor, inductor, and
capacitor are connected in series
with an ac source, the voltage and
current amplitudes are related by:
• The impedance of the circuit is:
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The L-R-C series circuit: Slide 2 of 3
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The L-R-C series circuit: Slide 3 of 3
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Measuring body fat by bioelectric impedance
analysis
• The electrodes attached to this
overweight patient’s chest are
applying a small ac voltage of
frequency 50 kHz.
• The attached instrumentation
measures the amplitude and phase
angle of the resulting current
through the patient’s body.
• These depend on the relative
amounts of water and fat along the path followed by the
current, and so provide a sensitive measure of body
composition.
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Power in a resistor
• If the circuit element
is a pure resistor, the
voltage and current
are in phase.
• The instantaneous
power p = vi is
always positive.
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Power in an inductor
• If the circuit
element is a pure
inductor, the
voltage leads the
current by 90°.
• The power is
negative when v
and i have opposite
signs, and positive
when they have the
same signs.
• The average power
is zero.
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Power in a capacitor
• If the circuit
element is a pure
capacitor, the
voltage lags the
current by 90°.
• The power is
negative when v
and i have opposite
signs, and positive
when they have the
same signs.
• The average power
is zero.
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Power in a general ac circuit
• For an arbitrary
combination of
resistors,
inductors, and
capacitors, the
average power is
positive.
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Power in a general ac circuit
• In any ac circuit, with any
combination of resistors, capacitors,
and inductors, the voltage v across
the entire circuit has some phase
angle ϕ with respect to the current i.
• The factor cos ϕ is called the
power factor of the circuit.
• For a pure resistor, the power
factor is 1.
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Resonance in ac circuits
• Shown are graphs of R, XL,
XC, and Z as functions of
log ω.
• As the frequency increases,
XL increases and XC
decreases; hence there is
always one frequency at
which XL and XC are equal
and XL − XC is zero.
• At this frequency the
impedance Z has its smallest
value, equal simply to the resistance R.
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Resonance in ac circuits
• As we vary the angular frequency ω of the source, the
maximum value of I occurs at the frequency at which the
impedance Z is minimum.
• This peaking of the current amplitude at a certain frequency
is called resonance.
• The angular frequency ω0 at which the resonance peak occurs
is called the resonance angular frequency.
• At ω = ω0 the inductive reactance XL and capacitive reactance
XC are equal, so ω0 L = 1/ω0 C and:
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Resonance in ac circuits
• Shown is a graph of current amplitude I as a function of
angular frequency ω for an L-R-C series circuit with
V = 100 V, L = 2.0 H, C = 0.50 mF, and three different values
of the resistance R.
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Transformers
• In a transformer,
power is supplied to a
primary coil, and then
the secondary coil
delivers power to a
resistor.
• The purpose of a stepup transformer, such
as the one shown, is to
increase the delivered
voltage relative to the
supplied voltage.
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Transformers
• In an ideal transformer, the ratio of the voltages across the
primary and secondary coils is equal to the ratio of the
number of turns in the coils:
• If N2 > N1, then V2 > V1 and we have a step-up transformer.
• If N2 < N1, then V2 < V1 and we have a step-down
transformer.
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